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Sliding Vectors, Line Bivectors, and Torque

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Sliding Vectors, Line Bivectors, and Torque Sliding vectors, line bivectors, and torque William G. Faris Department of Mathematics University of Arizona September 6, 2017 Abstract This paper is a modern exposition of old ideas. The setting is a Euclid- ian space E of dimension n with associated vector space V of dimension n. A (non-zero) sliding vector is a vector in V that is free to move, but only within a line L of E. The set of sliding vectors has dimension 2n − 1. n+1 This set is naturally embedded in a vector space of dimension 2 . An element of this vector space will be called a line bivector. Other terms used in applications are screw and wrench. There is a nice description of line bivectors in terms of Grassmann algebra in a projective representa- tion. It is shown that this abstract description has a concrete realization in terms of moment functions from E to bivectors over V . The literature in physics and engineering mainly deals with the special case n = 3. The results of the paper apply in this case and to its most common application, where the vectors in V represent force and the bivectors over V represent torque. It concludes with a discussion of duality, such as that of force and velocity or of torque and angular velocity. 1 Introduction Sliding vectors are vectors with a line of application. The vector is pictured as an arrow that is free to slide within its line. The space of sliding vectors is not closed under addition, but sliding vectors are included in a larger vector space. The purpose of this note is to give a brief description of the theory of these objects. Let E be n-dimensional Euclidean space, and let V be the associated n- dimensional vector space. There are three kinds of vectors. • A bound vector is a pair (P; u), where P is in E and u is in V . • A non-zero sliding vector is a pair (L; u) consisting of a line L in E together with a non-zero vector u in V that leaves L invariant. There is also a zero sliding vector (E; 0). • A free vector is a vector u in V . 1 Each bound vector P; u determines a sliding vector P ^ u. If u 6= 0, then the line L is determined by P and u. If Q is another point on the same line L, then P ^ u = Q ^ u. In the case when u = 0, every point P in E gives the zero sliding vector P ^ 0. Each sliding vector P ^ u determines a free vector u. The maps are summa- rized by (P; u) 7! P ^ u 7! u: (1) In the following the space of sliding vectors will be denoted E k V . It may be thought of as the space of all pairs (P; u) with P in E and u in V , subject to a certain equivalence relation. Thus (P; u) is equivalent to (Q; v) if u = v and the vectors Q − P and u are linearly dependent. With this notation, the maps given above send E × V ! E k V ! V . The dimension of the space of bound vectors is 2n, while the dimension of the space of free vectors is n. The dimension of the set of sliding vectors is 2n − 1. This may be seen by noting that for each non-zero free vector u the space of lines L that are aligned with u has dimension n − 1. Consider two non-zero sliding vectors P ^ u and Q ^ v with lines L and N. Suppose that L and N intersect in a point R. Then P ^ u = R ^ u and Q ^ v = R ^ v. It is natural to define the sum P ^ u + Q ^ v = R ^ u + R ^ v = R ^ (u + v): (2) Suppose that P ^ u and Q ^ u are non-zero sliding vectors with lines L and N that are parallel. Furthermore, suppose that u + v 6= 0. Then u = a(u + v) and v = b(u + v) with a + b = 1. It is natural to define the sum P ^ u + Q ^ v = P ^ a(u + v) + Q ^ b(u + v) = (aP + bQ) ^ (u + v): (3) Here aP + bQ is the weighted combination of points P; Q. Since a + b = 1 this is another point. When n ≥ 2 the sliding vectors do not form a vector space; the sum of sliding vectors need not be a sliding vector. When n = 2 there is only one way this can happen. This is with two lines L; N that are parallel and not equal and with non-zero vectors u; v with sum u + v = 0. When n ≥ 3 there are many pairs of lines that are not in the same plane, so it is very common for the sum of two sliding vectors not to be another sliding vector. Sliding vectors form part of a larger vector space. An element of this larger space will be called a line bivector or a screw. We shall see that the term \line bivector" is geometrically natural. Other terms like \screw" and \wrench" may be appropriate in physical applications. The precise definition of line bivector is given later on, but here is a brief preview. A line bivector may be represented (not uniquely) in the form M = P ^ u + α; (4) where α is a bivector built over the n dimensional vector space V . The space of n bivectors over V has dimension 2 . The dimension of the space of line bivectors n+1 is 2 . The collection of all sliding vectors P ^ u where P and u both vary is 2 not a vector space; it is a subset of the vector space of line bivectors of dimension 2n − 1. This is part of an old subject; the treatise that is often cited is Robert S. Ball, The Theory of Screws: A study in the dynamics of a rigid body, published in 1876. However the topic is still of current interest, in particular as a tool for robotics. Much of the work is in dimension n = 3, where it is natural to call a line bivector a line pseudovector. Books on statics (for instance [6]) often realize line pseudovectors as vector-pseudovector pairs that depend on a choice of reference point. A recent paper by Minguzzi [5] on screw theory treats line pseudovectors (screws) as moment functions. The moment function approach avoids the necessity of making an arbitrary choice of reference point. It also justifies the terminology used in the present paper; a point vector is a moment function on E whose values are vectors in V , while a line bivector is a moment function on E whose values are bivectors over V . Furthermore, a point vector typically defines a point in E, while a line bivector typically defines a line in E (the principal axis). The Minguzzi article gives other useful background information and is an excellent reference overall. The present treatment is for n dimensions. This was inspired by the book by Browne [2] on Grassmann algebra. This book presents a projective space point of view. Our treatment of line bivectors begins with this projective space picture. Then it is shown how the moment function description is a concrete realization of the projective space picture. The next part of the paper deals with situations where one makes use of the scalar product on the vector space V . In that case one can interpret bivectors as elements of a Lie algebra, more specifically, as infinitesimal rotations. The scalar product also gives a canonical form for line bivectors, the Poinsot central axis theorem. It also gives a way to describe the situation in dimension n = 3, where a bivector is often represented by a pseudovector. This part concludes with a brief sketch of the application to rigid body mechanics. The vector u is a force vector, and the line bivector P ^u+α represents a moment or torque about some unspecified point. The calculations begin and end with sliding vectors, but the intermediate steps use the more general line bivectors. The paper concludes with a discussion of a relation between line bivectors and certain dual objects. For the convenience of the reader, here is a summary list of notations. These are explained in the body of the paper. The list also includes some terms used in physical applications. • point P in vector space E of dimension n • vector u in vector space V of dimension n (force) • sliding vector P ^ u in space E k V of dimension 2n − 1 2 n • bivector α in vector space Λ (V ) of dimension 2 (couple, moment, torque) • point vector tP + u in vector space V • of dimension n + 1 3 2 • n+1 • line bivector P ^ u + α in vector space Λ (V ) of dimension 2 (screw, moment function, wrench) There are various relations between these spaces: • V ! V • is an injection. • E ! V • is an injection. • (P; u) 7! P ^ u is a mapping E × V ! E k V . • E k V ! Λ2(V •) is an injection. • Λ2(V ) ! Λ2(V •) is an injection. • P ^ u + α 7! u is a mapping Λ2(V •) ! V . Furthermore, for every point O∗ in E there is a corresponding isomorphism Λ2(V •) ! V ⊕ Λ2(V ). given by P ^ u + α 7! (u; (P − O∗) ^ u + α). 2 Sliding vectors in the plane The case n = 2 of sliding vectors in the plane is particularly simple. In rigid body mechanics each vector u represents a force.
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